Applying Generalized Rough Set Concepts to Approximation Spaces of Semigroups

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Applying Generalized Rough Set Concepts to

Approximation Spaces of Semigroups

Rukchart Prasertpong, Manoj Siripitukdet

Abstract—This paper presents generalized rough sets in approximation spaces based on portions of successor classes induced by arbitrary binary relations between two universes. Based on this definition, some interesting properties are in-vestigated. In semigroup structures, the notions of rough semigroups, rough ideals and rough completely prime ideals in approximation spaces induced by preorder and compatible relations are proposed. Some related results and examples are discussed and provided. In the end, the relationships between rough semigroups (resp. rough ideals and rough completely prime ideals) and their homomorphic images are verified. These associations are presented in understanding of necessary and sufficient conditions.

Index Terms—generalized rough set, approximation space, semigroup, rough semigroup, rough ideal, rough completely prime ideal, binary relation, preorder relation, compatible relation.

I. INTRODUCTION

T

HE Pawlak’s rough set theory is a influential tool for several assessment and decision problems of uncertain data in information technology. This classical theory was introduced by Pawlak [1]. The Pawlak’s rough set has been regarded as an approximation processing model in an approximation space induced by an equivalence relation on the single universe. For a given equivalence relation on a universal set and given a non-empty subset of the universal set, the Pawlak’s rough set of the given set is refered to as a pair of the Pawlak’s upper and Pawlak’s lower approximations where the difference between the Pawlak’s upper and Pawlak’s lower approximations (The Pawlak’s boundary region) is a non-empty set. The Pawlak’s upper approximation is the union of all the equivalence classes which have a non-empty intersection with the given set. The Pawlak’s lower approximation is the union of all the equivalence classes which are subset of the given set. As mentioned above, the Pawlak’s rough set model is referred to as a mathematical tool for an assessment and decision space in several information systems, including algebraic systems [2]–[15], expert systems with applications [16], knowledge-based information systems [17], computers and engineerings [18], measurements [19], approximate reasonings [20] etc. under intelligent information fields of artificial intelligence.

This research was partially funded by the Faculty of Science and Technology, Nakhon Sawan Rajabhat University, Nakhon Sawan 60000, Thailand and the Research Center for Academic Excellence in Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand.

R. Prasertpong is with the Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand (e-mail: [email protected]).

M. Siripitukdet is with the Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand and the Research Center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok 65000, Thailand. (corresponding author to provide e-mail: [email protected]).

Based on the notion of the Pawlak’s roughness model induced by an equivalence relation, generalizations of such the model have been being constructed by many researchers. Particularly, a generalization of Pawlak’s rough set based on an arbitrary binary relation (briefly, binary relation) on the single universe. In 1998, Yao [22] introduced a Yao’s rough set based on successor neighborhoods induced by a binary relation [SNθ(u) := {u′ ∈ U : (u, u′) ∈ θ} denotes a

successor neighborhood of u induced by a binary relation θ on a universal set U where u is an element in U]. In 2016, Mareay [23] introduced a new rough set by using cores of successor neighborhoods induced by a binary relation [CSNθ(u) := {u′ ∈ U : SNθ(u) = SNθ(u′)} denotes a

core of a successor neighborhood ofuinduced by a binary relationθon a universal setU whereuis an element inU]. Observer that the Yao’s rough set and the Mareay’s rough set are generalizations of the Pawlak’s rough set whenever the binary relation is an equivalence relation.

The semigroup structure (see [24]) is an algebraic sys-tem with respect to wide applications. For employments of Pawlak’s rough set theory in semigroup, Kuroki [4] introduced the notions of upper and lower approximation semigroups (resp. ideals) in semigroups induced by congru-ence relations, and provided sufficient conditions of upper and lower approximation semigroups (resp. ideals) in 1997. In 2006, Xiao and Zhang [7] introduced the notions of upper and lower approximation completely prime ideals in semigroups induced by congruence relations, and provided sufficient conditions of upper and lower approximation com-pletely prime ideals. They examined the relationship between upper and lower approximation completely prime ideals (resp. ideals) and their homomorphic images under homo-morphism problems. In 2016, Wang and Zhan [13] proposed the notions of upper and lower approximation semigroups (resp. ideals and completely prime ideals) induced by specific congruence relations, and also provided sufficient conditions of upper and lower approximation semigroups (resp. ideals and completely prime ideals).

Based on the generalized rough set in sense of Mareay, the main point of this work is a extended concept, i.e., a general-ization of the Mareay’s rough set induced by a binary relation between two universes will be established. Then we apply the novel rough set in semigroups for approximation processings. After providing some fundamentals of binary relations and semigroups in Section II, we propose a generalization of the Mareay’s rough set in an approximation space of a universal set based on portions of successor classes induced by a binary relation between two universes and investigate some interesting properties in Section III. In Section IV, we introduce concepts of rough semigroups, rough ideals and rough completely prime ideals in approximation spaces of semigroups under preorder and compatible relations. Next,

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we provide sufficient conditions and examples of them. In Section V, we verify relationships between rough semigroups (resp. rough ideals and rough completely prime ideals) and their homomorphic images. Lastly, this paper is concluded with some remarks and discussions in Section VI.

II. PRELIMINARIES

In this section we reconsider some important definitions which will be employed in the following sections.

Throughout this paper, U and V denote two non-empty universe.

Definition 1. [21] LetΞ(U×V)be a family of all subsets ofU×V. An element inΞ(U×V)is referred to as abinary relation fromU toV. An element inΞ(U×V)is called a

binary relation on U ifU =V.

Definition 2. [21] LetΛbe a binary relation fromU toV. Λ is called serial if for allu∈U, there existsv ∈V such that (u, v)∈Λ.

Definition 3. [21] LetΛbe a binary relation on U. (1) Λ is calledreflexiveif for allu∈U,(u, u)∈Λ. (2) Λis calledtransitiveif for allu1, u2, u3∈U,(u1, u2)∈

Λ and(u2, u3)∈Λimplies (u1, u3)∈Λ.

(3) Λis calledsymmetricif for allu1, u2∈U,(u1, u2)∈Λ

implies(u2, u1)∈Λ.

(4) If Λ is reflexive and transitive, then Λ is called a

preorder.

(5) If Λ is reflexive, transitive and symmetric, then Λ is called anequivalence relation.

Asemigroup[24](S,⊙)is defined as an algebraic system whereS is a non-empty set and ⊙is an associative binary operation on S. Throughout this paper, S denotes a semi-group. A non-empty subsetX ofS is called asubsemigroup

[25] ofS ifXX⊆X. A non-empty subsetX ofS is called a left (right) ideal [25] of S if SX ⊆ X (XS ⊆X), and if it is both a left ideal and a right ideal of S, then it is called anideal[25]. An idealX ofS is called acompletely prime ideal[25] ofS if for alls1, s2∈S,s1s2∈X implies

s1 ∈X or s2∈X. LetΛ be a binary relation on S. Then,

Λ is calledcompatible if for alls1, s2, s3∈S,(s1, s2)∈Λ

implies(s1s3, s2s3)∈Λand(s3s1, s3s2)∈Λ.

III. GENERALIZED ROUGH SETS

In this section we establish a generalization of the Mareay’s rough set induced by a binary relation between two universes and provide a real-world example and verify some appealing properties.

Definition 4. Let Λ be a binary relation fromU toV. For an elementu∈U,

SΛ(u) :={v∈V : (u, v)∈Λ}

is called asuccessor class ofuinduced by Λ.

Remark 1. If Λ is a serial relation from U to V, then SΛ(u)̸=∅for allu∈U.

Definition 5. Let Λ be a binary relation fromU toV. For an elementu1∈U,

P SΛ(u1) :={u2∈U :SΛ(u2)⊆SΛ(u1)}

is called aportion of the successor class of u1 induced by

Λ.

We denote by PSΛ(U) a collection of P SΛ(u) for all

u∈U.

Directly from Definition 5, we can obtain to Proposition 1 as the following.

Proposition 1. LetΛbe a binary relation fromUtoV. Then the following statements hold.

(1) For allu∈U,u∈P SΛ(u).

(2) For all u1, u2 ∈ U, u1 ∈ P SΛ(u2) if and only if

P SΛ(u1)⊆P SΛ(u2).

Proposition 2. Let Λ be a binary relation on U. Then we have the following statements.

(1) IfΛ is reflexive, thenP SΛ(u)⊆SΛ(u)for all u∈U. (2) IfΛis transitive, thenSΛ(u)⊆P SΛ(u)for allu∈U. (3) IfΛis a preorder, thenSΛ(u)andP SΛ(u)are identical

classes for allu∈U.

Proof:(1) It is clear thatP SΛ(u)⊆SΛ(u)for allu∈U

wheneverΛ is reflexive.

(2) Letu1∈U and letu2∈SΛ(u1). Then,(u1, u2)∈Λ.

We only need to show thatSΛ(u2)⊆SΛ(u1). Suppose that

u3 ∈ SΛ(u2). Then, (u2, u3) ∈ Λ. Since Λ is transitive,

we have (u1, u3) ∈ Λ. Thus u3 ∈ SΛ(u1). Hence we get

SΛ(u2)⊆SΛ(u1). Therefore,u2∈P SΛ(u1). Consequently,

SΛ(u1)⊆P SΛ(u1).

(3) From items (1) and (2), it follows that this statement holds.

In the following, we give a generalization of the Mareay’s roughness model induced by a binary relation.

Definition 6. Let Λbe a binary relation from U toV. The triple (U, V,PSΛ(U)) is referred to as an approximation space based on PSΛ(U) (briefly, PSΛ(U)-approximation space). If U = V, then (U, V,PSΘ(U)) is replaced by a

pair(U,PSΘ(U)).

Definition 7. Let (U, V,PSΛ(U)) be an PSΛ(U)

-approximation space. For a non-empty subset X of U, we define three sets as follows:

Λ(X) :={u∈U :P SΛ(u)∩Xis a non-empty set},

Λ(X) :={u∈U :P SΛ(u)⊆X} and

Λbnd(X) :=Λ(X)−Λ(X).

Then

(1) Λ(X) denotes an upper approximation of X in

(U, V,PSΛ(U))(briefly,PSΛ(U)-upper approximation

ofX).

(2) Λ(X) denotes a lower approximation of X in

(U, V,PSΛ(U))(briefly,PSΛ(U)-lower approximation

ofX).

(3) Λbnd(X) denotes a boundary region of X in (U, V,PSΛ(U)) (briefly, PSΛ(U)-boundary region of

X).

(4) IfΛbnd(X)̸=∅, thenΛ(X) := (Λ(X), Λ(X))is called

arough set ofX in(U, V,PSΛ(U))(briefly,PSΛ(U)

-rough set ofX).

(5) If Λbnd(X) = ∅, then X is called a definable set in (U, V,PSΛ(U))(briefly,PSΛ(U)-definable set).

We give an example as the following.

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Example 1. LetU :={u1, u2, u3, u4, u5} be a set of

elec-trical discharge machines (EDM) in an automobile industry of leading company and letV :={v1, v2, v3, v4} be a set of

components of each elements inU.

Define TABLE I by an information of the damage values (Bad and Medium) of electrical discharge machines in U

with respect to components inV as the following detail.

TABLE I

THE INFORMATION TABLE OF DAMAGE VALUES

v1 v2 v3 v4 u1 Medium Bad Bad Medium u2 Medium Bad Bad Medium u3 Medium Bad Bad Bad u4 Medium Medium Medium Bad u5 Bad Bad Medium Medium

For a binary relationΛ∈Ξ(U×V)and elementsu∈U,

v ∈ V, a pair (u, v) ∈ Λ is defined as a bad damage value of the electrical discharge machine uwith respect to the componentvunderΛ. Then the binary relationΛis a set

{(u1, v2),(u1, v3),(u2, v2),(u2, v3),(u3, v2),(u3, v3),(u3, v4),

(u4, v4),(u5, v1),(u5, v2)}. Suppose that a measurement

expert committee assign X := {u1, u3, u5} which is a

non-empty set of electrical discharge machines for the discharge under a global evaluation. Then the assessment of X in approximation space (U, V,PSΛ(U)) is derived by a process as the following. According to Definition 4, it follows that

SΛ(u1) :={v2, v3},

SΛ(u2) :={v2, v3},

SΛ(u3) :={v2, v3, v4},

SΛ(u4) :={v4}and

SΛ(u5) :={v1, v2}.

According to Definition 5, it follows that

P SΛ(u1) :={u1, u2},

P SΛ(u2) :={u1, u2},

P SΛ(u3) :={u1, u2, u3, u4},

P SΛ(u4) :={u4}and

P SΛ(u5) :={u5}.

According to Definition 7, it follows that

Λ(X) :={u1, u2, u3, u5},

Λ(X) :={u5}and

Λbnd(X) :={u1, u2, u3}.

Therefore, Λ(X) := ({u1, u2, u3, u5},{u5}) is a PSΛ(U) -rough set ofX. As a consequence,

(1) u1, u2, u3 and u5 are possibly electrical discharge

ma-chines for the discharge,

(2) u5 is a certainly electrical discharge machine for the

discharge and

(3) u1, u2 and u3 cannot be determined whether three

machines are electrical discharge machines for the discharge or not.

The following remark is immediate consequences of Def-inition 7 and the existence of Example 1 with respect to Definition 7.

Remark 2. Every Mareay’s rough set in [23] is a rough set in Definition 7, but the converse is not true in general. Therefore, the rough set in Definition 7 is considered as a generalization of the Mareay’s rough set whenever U =V

and the relation “⊆” is substituted by the equality “ = ”

in Definition 5.

The existence of Example 1 leads to the following defini-tion.

-approximation space and letX be a non-empty subset ofU. Λ(X) is called a non-empty PSΛ(U)-upper approximation

of X in (U, V,PSΛ(U)) if Λ(X) is a non-empty subset

of U. Analogously, we can define non-empty PSΛ(U)

-lower approximations. ThePSΛ(U)-rough setΛ(X)ofX in (U, V,PSΛ(U))is referred to as anon-emptyPSΛ(U)-rough set if Λ(X) is a non-empty PSΛ(U)-upper approximation

andΛ(X)is a non-emptyPSΛ(U)-lower approximation.

Proposition 3. Let (U, V,PSΛ(U)) be an PSΛ(U) -approximation space. IfX and Y are non-empty subsets of

U, then we have the following statements.

(1) Λ(U) =U and Λ(U) =U. (2) Λ(∅) =∅ andΛ(∅) =∅. (3) X⊆Λ(X)and Λ(X)⊆X. (4) Λ(X∪Y) =Λ(X)∪Λ(Y)and

Λ(X∩Y) =Λ(X)∩Λ(Y). (5) Λ(X∩Y)⊆Λ(X)∩Λ(Y)and

Λ(X∪Y)⊇Λ(X)∪Λ(Y).

(6) IfX ⊆Y, thenΛ(X)⊆Λ(Y)and Λ(X)⊆Λ(Y).

Proof:(1)-(3) follow from Proposition 1 (1). The proofs (4), (5) and (6) are straightforward.

-approximation space and let X be a non-empty subset of U. If Λ(X) is a non-empty PSΛ(U)-lower approximation

ofX in(U, V,PSΛ(U))andΛ(X)is a proper subset ofX,

thenX is called aset over non-empty interior set.

Proposition 4. Let (U, V,PSΛ(U)) be an PSΛ(U) -approximation space and let X be a non-empty subset of

U. IfX is a set over non-empty interior set, then Λ(X)is a non-emptyPSΛ(U)-rough set ofX in(U, V,PSΛ(U)).

Proof:Suppose thatX is a set over non-empty interior set. Then we have thatΛ(X)is a non-emptyPSΛ(U)-lower

approximation and Λ(X) ⊂ X. By Proposition 3 (3), we obtain that X ⊆Λ(X). Thus we get Λ(X) is a non-empty PSΛ(U)-upper approximation. We shall verify thatΛbnd(X)

is a non-empty set. Suppose that Λbnd(X) = ∅. Then we

have Λ(X) = Λ(X). From Proposition 3 (3), once again, it follows that Λ(X) = X, which is a contradiction. Thus Λbnd(X)is a non-empty set. As a consequence, Λ(X)is a

non-emptyPSΛ(U)-rough set of X.

Proposition 5. Let (U,PSΛ(U)) be an PSΛ(U) -approximation space and let(U,PSΥ(U)) be anPSΥ(U) -approximation space. If Λ ⊆ Υ where Λ is reflexive and

Υ is transitive, then Λ(X) ⊆ Υ(X) for every non-empty subsetX ofU.

Proof: Let X be a non-empty subset of U. Then we prove thatΛ(X)⊆Υ(X). Infact, let u1 ∈Λ(X). Then we

have P SΛ(u1)∩X is a non-empty set. Thus there exists

u2 ∈ U such that u2 ∈ P SΛ(u1)∩X. Hence SΛ(u2) ⊆

SΛ(u1). SinceΛis reflexive, we have(u2, u2)∈Λ. Whence

we get that u2 ∈ SΛ(u2) ⊆ SΛ(u1). Thus (u1, u2) ∈ Λ.

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Since Λ⊆Υ,(u1, u2)∈Υ. We shall verify that SΥ(u2)⊆

SΥ(u1). Let now u3 ∈ SΥ(u2). Then, (u2, u3) ∈Υ. Since

Υ is a transitive relation, we have (u1, u3) ∈ Υ. Thus we

get u3 ∈ SΥ(u1), which yields SΥ(u2) ⊆ SΥ(u1). Hence

u2 ∈P SΥ(u1). Thus we have u2 ∈ P SΥ(u1)∩X. Hence

P SΥ(u1)∩X is a non-empty set, which yields u1∈Υ(X).

This implies that Λ(X)⊆Υ(X).

Proposition 6. Let (U,PSΛ(U)) be an PSΛ(U) -approximation space and let(U,PSΥ(U))be anPSΥ(U) -approximation space. If Λ ⊆ Υ where Λ is reflexive and

Υ is transitive, then Υ(X) ⊆ Λ(X) for every non-empty subset X ofU.

Proof: Let X be a non-empty subset of U. Then we prove thatΥ(X)⊆Λ(X). Indeed, suppose thatu1∈Υ(X).

Then we get that P SΥ(u1) ⊆ X. We shall show that

P SΛ(u1) ⊆ P SΥ(u1). Let u2 ∈ P SΛ(u1). Then we have

SΛ(u2) ⊆ SΛ(u1). Since Λ is a reflexive relation, we

have (u2, u2) ∈ Λ. Hence we get u2 ∈ SΛ(u2), and so

u2∈SΛ(u1). Thus we get(u1, u2)∈Λ. By the assumption,

we obtain that (u1, u2) ∈ Υ. Now, we shall prove that

SΥ(u2)⊆ SΥ(u1). Let u3 ∈ SΥ(u2). Then, (u2, u3) ∈Υ.

Since Υ is a transitive relation, we have (u1, u3) ∈ Υ.

Whence u3 ∈ SΥ(u1). Hence SΥ(u2) ⊆ SΥ(u1). Thus

u2∈P SΥ(u1). Whence we getP SΛ(u1)⊆P SΥ(u1)⊆X.

Therefore, u1 ∈Λ(X). As a consequence,Υ(X)⊆Λ(X).

IV. ROUGHNESS IN SEMIGROUPS

In this section we introduce rough semigroups, rough ideals and rough completely prime ideals in semigroups induced by preorder and compatible relations. Then we pro-vide sufficient conditions of them and give some interesting properties and examples.

Definition 10. Let (S,PSΛ(S)) be an PSΛ(S)

-approximation space. (S,PSΛ(S)) is called an PSΛ(S) -approximation space type PCR if Λ is a preorder and compatible relation.

Proposition 7. Let (S,PSΛ(S)) be an PSΛ(S) -approximation space type PCR. Then,

(P SΛ(s1))(P SΛ(s2))⊆P SΛ(s1s2)

for all s1, s2∈S.

Proof: Let s1 and s2 be two elements in S.

Sup-pose that s3 ∈ (P SΛ(s1))(P SΛ(s2)). Then there exist

s4 ∈ P SΛ(s1), s5 ∈ P SΛ(s2) such that s3 = s4s5.

Thus SΛ(s4) ⊆ SΛ(s1) and SΛ(s5) ⊆ SΛ(s2). Hence

we get that SΛ(s4s5) ⊆ SΛ(s1s2). Indeed, we suppose

that s6 ∈ SΛ(s4s5). Then, (s4s5, s6) ∈ Λ. Since Λ is

reflexive, we have (s4, s4) and (s5, s5) are in Λ, and so

s4 ∈ SΛ(s4) ands5 ∈ SΛ(s5). Whence s4 ∈ SΛ(s1) and

s5 ∈ SΛ(s2). Thus (s1, s4) ∈ Λ and (s2, s5) ∈ Λ. Since

Λ is compatible, we have (s1s2, s4s5) ∈ Λ. Since Λ is

transitive, we have (s1s2, s6) ∈ Λ. Whence we get that

s6∈SΛ(s1s2). Hence we obtain thatSΛ(s4s5)⊆SΛ(s1s2),

which yields s3 = s4s5 ∈ P SΛ(s1s2). As a consequence,

(P SΛ(s1))(P SΛ(s2))⊆P SΛ(s1s2).

We give to Example 2 as the following.

Example 2. Let S :={s1, s2, s3, s4, s5} be the semigroup

with multiplication rules defined by the TABLE II.

TABLE II

THE MULTIPLICATION TABLE ONS

· s1 s2 s3 s4 s5

s1 s1 s2 s3 s2 s5 s2 s2 s2 s3 s2 s5 s3 s3 s3 s3 s3 s3 s4 s2 s2 s3 s2 s5 s5 s5 s5 s3 s5 s5

DefineΛ:={(s1, s1),(s2, s2),(s2, s5),(s3, s2),(s3, s3),

(s3, s5),(s4, s4),(s5, s2),(s5, s5)}. Then it is easy to check

thatΛis a preorder and compatible relation. Thus successor classes of each elements inS induced byΛ are as follows:

SΛ(s1) :={s1},

SΛ(s2) :={s2, s5},

SΛ(s3) :={s2, s3, s5},

SΛ(s4) :={s4}and

SΛ(s5) :={s2, s5}.

Hence by Proposition 2 (3), we obtain that

P SΛ(s1) =SΛ(s1),

P SΛ(s2) =SΛ(s2),

P SΛ(s3) =SΛ(s3),

P SΛ(s4) =SΛ(s4)and

P SΛ(s5) =SΛ(s5).

Here it is straightforward to verify that for alls, s′∈S

(P SΛ(s))(P SΛ(s′))⊆P SΛ(ss′).

Observe that, in Example 2, it does not holds in general for an equality case. We consider the following example.

Example 3. Let S :={s1, s2, s3, s4, s5} be the semigroup

with multiplication rules defined by the TABLE III.

TABLE III

THE MULTIPLICATION TABLE ONS

· s1 s2 s3 s4 s5

s1 s1 s1 s3 s1 s5 s2 s1 s2 s3 s1 s5 s3 s3 s3 s3 s3 s3 s4 s1 s1 s3 s4 s5 s5 s5 s5 s3 s5 s5

DefineΛ:={(s1, s1),(s1, s2),(s1, s4),(s2, s1),(s2, s2),

(s2, s4),(s3, s3),(s3, s5),(s4, s1),(s4, s2),(s4, s4),(s5, s5)}.

Then it is easy to check thatΛis a preorder and compatible relation. Thus successor classes of each elements in S

induced byΛare as follows:

SΛ(s1) :={s1, s2, s4},

SΛ(s2) :={s1, s2, s4},

SΛ(s3) :={s3, s5},

SΛ(s4) :={s1, s2, s4} and

SΛ(s5) :={s5}.

Thus by Proposition 2 (3), we obtain that

P SΛ(s1) =SΛ(s1),

P SΛ(s2) =SΛ(s2),

P SΛ(s3) =SΛ(s3),

P SΛ(s4) =SΛ(s4)and

P SΛ(s5) =SΛ(s5).

Here it is straightforward to check that for alls, s′∈S

(P SΛ(s))(P SΛ(s′)) =P SΛ(ss′).

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Considering this point, the property can be considered as a special case of Proposition 7. This important example leads to Definition 11 as the following.

-approximation space type PCR. The collection PSΛ(S)

is called a complete collection induced by Λ (briefly, Λ-complete) if for alls1, s2∈S,

(P SΛ(s1))(P SΛ(s2)) =P SΛ(s1s2).

-approximation space type PCR. If PSΛ(S) is a complete

collection induced by Λ, then Λ is called a complete relation. (S,PSΛ(S)) is called an PSΛ(S)-approximation space type CR ifΛ is complete.

Proposition 8. Let (S,PSΛ(S)) be an PSΛ(S) -approximation space type PCR. Then,

(Λ(X))(Λ(Y))⊆Λ(XY)

for every non-empty subsetsX, Y ofS.

Proof:LetX, Y be two non-empty subsets ofS and let s1∈(Λ(X))(Λ(Y)). Then there existss2∈Λ(X)and exists

s3 ∈ Λ(Y) such that s1 = s2s3. Thus we get P SΛ(s2)∩

X and P SΛ(s3)∩Y are non-empty sets. Then there exist

s4, s5∈Ssuch thats4∈P SΛ(s2)∩X ands5∈P SΛ(s3)∩

Y. From Proposition 7, it follows that

s4s5∈(P SΛ(s2))(P SΛ(s3))⊆P SΛ(s2s3).

Note that s4s5 ∈ XY. Thus P SΛ(s2s3)∩XY is a

non-empty set, which yields s1=s2s3∈Λ(XY). Therefore we

get (Λ(X))(Λ(Y))⊆Λ(XY).

Proposition 9. Let (S,PSΛ(S)) be an PSΛ(S) -approximation space type CR. Then,

(Λ(X))(Λ(Y))⊆Λ(XY)

for every non-empty subsetsX, Y ofS.

Proof: Let X and Y be two non-empty subsets of S and let s1 ∈ (Λ(X))(Λ(Y)). Then there exist s2 ∈ Λ(X)

and s3 ∈ Λ(Y) such that s1 = s2s3. Hence we get that

P SΛ(s2)⊆X andP SΛ(s3)⊆Y. SinceΛ is complete,

P SΛ(s2s3) =P SΛ(s2)P SΛ(s3)⊆XY.

Whence we obtain P SΛ(s2s3) ⊆ XY. Hence we get that

s1=s2s3∈Λ(XY). Therefore,(Λ(X))(Λ(Y))⊆Λ(XY).

In what follows, a rough set in semigroups will be pro-posed. We consider to Example 4 as the following.

Example 4. According to Example 3, we let X :=

{s2, s3, s5} be a subset ofS. Then we haveΛ(X) =S and

Λ(X) :={s3, s5}. Here it is easy to verify that Λ(X)and

Λ(X)are subsemigroups, ideals and completely prime ideals ofS. Moreover, we also have Λbnd(X)is a non-empty set. Existences of subsemigroups, ideals and completely prime ideals ofS induced by preorder and compatible relations in this example lead to Definition 13 as the following.

-approximation space type PCR and let X be a non-empty subset of S. The non-empty PSΛ(S)-upper approximation

Λ(X) of X in (S,PSΛ(S)) is called an PSΛ(S)-upper approximation semigroup if it is a subsemigroup of S. The non-empty PSΛ(S)-lower approximation Λ(X) of X

in (S,PSΛ(S)) is called a PSΛ(S)-lower approximation semigroup if it is a subsemigroup of S. The non-empty PSΛ(S)-rough set Λ(X) of X in (S,PSΛ(S)) is called

a PSΛ(S)-rough semigroup if Λ(X) is an PSΛ(S)-upper

approximation semigroup and Λ(X) is a PSΛ(S)-lower

approximation semigroup. Similarly, we can definePSΛ(S)

-rough (completely prime) ideals.

Theorem 1. Let(S,PSΛ(S))be anPSΛ(S)-approximation space type PCR. IfX is a subsemigroup ofS, thenΛ(X)is anPSΛ(S)-upper approximation semigroup.

Proof: Suppose thatX is a subsemigroup ofS. Then, XX⊆X. By Proposition 3 (3), we obtain that

∅ ̸=X⊆Λ(X).

Hence we getΛ(X)is a non-emptyPSΛ(S)-upper

approx-imation. From Proposition 3 (6), it follows thatΛ(XX)⊆ Λ(X). By Proposition 8, we get

(Λ(X))(Λ(X))⊆Λ(XX)⊆Λ(X).

Hence Λ(X) is a subsemigroup of S. Thus Λ(X) is an PSΛ(S)-upper approximation semigroup.

Theorem 2. Let(S,PSΛ(S))be anPSΛ(S)-approximation space type CR. IfX is a subsemigroup ofS withΛ(X)is a non-empty set, thenΛ(X)is aPSΛ(S)-lower approximation semigroup.

Proof: Suppose thatX is a subsemigroup ofS. Then, XX ⊆ X. Obviously, Λ(X) is a non-empty PSΛ(S)

-lower approximation. From Proposition 3 (6), it follows that Λ(XX)⊆Λ(X). By Proposition 9, we obtain that

(Λ(X))(Λ(X))⊆Λ(XX)⊆Λ(X).

Thus Λ(X) is a subsemigroup of S. Therefore, Λ(X) is a PSΛ(S)-lower approximation semigroup.

The following corollary is immediate consequences of Proposition 4, Theorem 1 and Theorem 2.

Corollary 1. Let(S,PSΛ(S))be anPSΛ(S)-approximation space type CR. IfX is a subsemigroup ofSover non-empty interior set, thenΛ(X)is aPSΛ(S)-rough semigroup.

Observe that, in Corollary 1, the converse is not true in general. We present an example as the following.

Example 5. According to Example 3, suppose that X :=

{s2, s4, s5} is a subset of S, then we have that Λ(X) :=

{s1, s2, s4, s5}and Λ(X) :={s5}. ThusΛbnd(X)is a non-empty set. Hence it is straightforward to check that Λ(X)

is anPSΛ(S)-upper approximation semigroup andΛ(X)is a PSΛ(S)-lower approximation semigroup. However, X is not a subsemigroup ofS. Consequently,Λ(X)is aPSΛ(S) -rough semigroup, butX is not a subsemigroup of S.

Theorem 3. Let(S,PSΛ(S))be anPSΛ(S)-approximation space type PCR. If X is an ideal of S, then Λ(X) is an

PSΛ(S)-upper approximation ideal.

Proof:Suppose that X is an ideal ofS. Then we have SX⊆X. From Proposition 3 (6), it follows thatΛ(SX)⊆

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Λ(X). By Proposition 3 (1), we obtain thatΛ(S) =S. From Proposition 8, it follows that

S(Λ(X)) = (Λ(S))(Λ(X))⊆Λ(SX)⊆Λ(X).

Hence Λ(X)is a left ideal of S.

Similarly, we can prove that Λ(X)is a right ideal of S. Therefore, Λ(X) is an PSΛ(S)-upper approximation ideal.

Theorem 4. Let(S,PSΛ(S))be anPSΛ(S)-approximation space type CR. If X is an ideal of S withΛ(X)is a non-empty set, then Λ(X) is a PSΛ(S)-lower approximation ideal.

Proof:Suppose thatXis an ideal ofS. Then,SX ⊆X. From Proposition 3 (6), we getΛ(SX)⊆Λ(X). By Propo-sition 3 (1), we obtain that Λ(S) =S. From Proposition 9, it follows that

S(Λ(X)) = (Λ(S))(Λ(X))⊆Λ(SX)⊆Λ(X).

ThusΛ(X)is a left ideal ofS.

Similarly, we can prove that Λ(X)is a right ideal of S. ThusΛ(X)is aPSΛ(S)-lower approximation ideal.

Corollary 2. Let(S,PSΛ(S))be anPSΛ(S)-approximation space type CR. IfX is an ideal ofSover non-empty interior set, then Λ(X) is aPSΛ(S)-rough ideal.

Example 6. According to Example 3, if X :={s3, s4, s5}

is a subset of S, then we have Λ(X) = S and Λ(X) :=

{s3, s5}. Thus we getΛbnd(X)is a non-empty set. Obviously,

Λ(X) is an PSΛ(S)-upper approximation ideal, and it is straightforward to check that Λ(X) is a PSΛ(S)-lower approximation ideal. However, X is not an ideal of S. Consequently,Λ(X)is aPSΛ(S)-rough ideal, butX is not an ideal ofS.

Theorem 5. Let(S,PSΛ(S))be anPSΛ(S)-approximation space type CR. If X is a completely prime ideal ofS, then

Λ(X)is anPSΛ(S)-upper approximation completely prime ideal.

Proof: Suppose that X is a completely prime ideal of S. Then we prove thatΛ(X)is anPSΛ(S)-upper

approxi-mation completely prime ideal. In fact, since X is an ideal of S, by Theorem 3, we have that Λ(X) is an PSΛ(S)

-upper approximation ideal. Let s1, s2 ∈ S be such that

s1s2∈Λ(X). Then, by theΛ-complete property ofPSΛ(S),

we get that

(P SΛ(s1))(P SΛ(s2))∩X =P SΛ(s1s2)∩X

is a non-empty set. Thus there exist s3 ∈ P SΛ(s1), s4 ∈

P SΛ(s2)such thats3s4∈X. SinceXis a completely prime

ideal, s3 ∈ X or s4 ∈ X. Thus P SΛ(s1)∩X is a

non-empty set or P SΛ(s2)∩X is a non-empty set. Hence s1∈

Λ(X)ors2∈Λ(X). Therefore,Λ(X)is a completely prime

ideal of S. As a consequence, Λ(X) is an PSΛ(S)-upper

approximation completely prime ideal.

Theorem 6. Let(S,PSΛ(S))be anPSΛ(S)-approximation space type CR. If X is a completely prime ideal of S with

Λ(X) is a non-empty set, then Λ(X) is a PSΛ(S)-lower approximation completely prime ideal.

Proof: Suppose that X is a completely prime ideal of S with Λ(X) ̸= ∅. Then, X is an ideal of S. Thus by Theorem 4, we haveΛ(X)is aPSΛ(S)-lower approximation

ideal. Lets1, s2∈S be such thats1s2∈Λ(X). Since Λis

complete, we have

(P SΛ(s1))(P SΛ(s2)) =P SΛ(s1s2)⊆X.

Now, we suppose thats1 ∈/ Λ(X). Then we have P SΛ(s1)

is not a subset of X. Thus there exists s3 ∈ S such that

s3∈P SΛ(s1)buts3∈/X. For eachs4∈P SΛ(s2),

s3s4∈(P SΛ(s1))(P SΛ(s2))⊆X.

Whence s3s4 ∈ X. Since X is a completely prime ideal

ands3∈/ X, we haves4∈X. Thus we getP SΛ(s2)⊆X,

which yieldss2∈Λ(X). Hence we getΛ(X)is a completely

prime ideal of S. Therefore, Λ(X) is a PSΛ(S)-lower

approximation completely prime ideal.

Corollary 3. Let(S,PSΛ(S))be anPSΛ(S)-approximation space type CR. If X is a completely prime ideal of S

over non-empty interior set, thenΛ(X)is aPSΛ(S)-rough completely prime.

Example 7. According to Example 3, if X :={s1, s3, s5}

is a subset of S, then we have Λ(X) = S and Λ(X) :=

{s3, s5}. Hence we getΛbnd(X)̸=∅. Obviously,Λ(X)is an

PSΛ(S)-upper approximation completely prime ideal, and it is straightforward to check thatΛ(X)is aPSΛ(S)-lower approximation completely prime ideal. Here we can verify that X is an ideal of S, but it is not a completely prime ideal ofSsinces2s4=s1∈X buts2∈/X ands4∈/ X. As

a consequence,Λ(X)is aPSΛ(S)-rough completely prime ideal, butX is not a completely prime ideal of S.

V. HOMOMORPHIC IMAGES OF ROUGHNESS IN SEMIGROUPS

In this section we verify relationships between rough semi-groups (resp. rough ideals, rough completely prime ideals) and their homomorphic images. Throughout this section, we suppose thatT denotes a semigroup.

Proposition 10. Let f be an epimorphism from S in

(S,PSΛ(S))toTin(T,PSΥ(T)), where the binary relation

Λ := {(s1, s2) ∈ S×S : (f(s1), f(s2)) ∈ Υ}. Then the

following statements hold.

(1) For alls1, s2∈S,s1∈P SΛ(s2)if and only iff(s1)∈

P SΥ(f(s2)).

(2) f(Λ(X)) =Υ(f(X))for every non-empty subsetX of

S.

(3) f(Λ(X))⊆Υ(f(X))for every non-empty subsetX of

S.

(4) If f is injective, then f(Λ(X)) = Υ(f(X)) for every non-empty subsetX ofS.

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(5) IfΥ is a preorder and compatible relation, thenΛis a preorder and compatible relation.

Proof: (1) Let s1, s2∈ S be such that s1 ∈P SΛ(s2).

Then, f(s1), f(s2) ∈ T and SΛ(s1) ⊆ SΛ(s2). We shall

prove that SΥ(f(s1)) ⊆ SΥ(f(s2)). Let t1 ∈ SΥ(f(s1)).

Then, (f(s1), t1) ∈ Υ. Since f is surjective, there exists

s3 ∈ S such thatf(s3) =t1. Whence (f(s1), f(s3))∈Υ,

and so (s1, s3) ∈ Λ. Thus s3 ∈ SΛ(s1). Whence we

have s3 ∈ SΛ(s2). Hence we have (s2, s3) ∈ Λ, and

so (f(s2), f(s3)) ∈ Υ. Thus t1 = f(s3) ∈ SΥ(f(s2)).

Then we have SΥ(f(s1)) ⊆ SΥ(f(s2)). Therefore we get

f(s1)∈P SΥ(f(s2)).

Conversely, it is easy to verify that s1 ∈ P SΛ(s2)

wheneverf(s1)∈P SΥ(f(s2))for alls1, s2∈S.

(2) Let X be a non-empty subset of S. We verify firstly that f(Λ(X)) = Υ(f(X)). Let t1 ∈ f(Λ(X)). Then

there exists s1 ∈ Λ(X) such that f(s1) = t1. Therefore,

P SΛ(s1)∩X ̸= ∅. Thus there exists s2 ∈ S such that

s2 ∈ P SΛ(s1) and s2 ∈ X. By item (1), we obtain

that f(s2) ∈ P SΥ(f(s1)) and f(s2) ∈ f(X). Then,

P SΥ(f(s1))∩f(X)̸= ∅, and so t1 = f(s1) ∈ Υ(f(X)).

Thus we have f(Λ(X))⊆Υ(f(X)).

On the other hand, we let t2 ∈ Υ(f(X)). Then there

exists s3 ∈ S such that f(s3) = t2. Hence we get that

P SΥ(f(s3))∩f(X)̸=∅. Thus there existss4∈X such that

f(s4)∈f(X)and f(s4) ∈P SΥ(f(s3)). By the argument

(1), we get that s4 ∈P SΛ(s3), and so P SΛ(s3)∩X ̸=∅.

Hence s3 ∈ Λ(X), and so t2 = f(s3) ∈ f(Λ(X)). Thus

Υ(f(X))⊆f(Λ(X)). Therefore,f(Λ(X)) =Υ(f(X)). (3) Let X be a non-empty subset of S. Suppose that t1 ∈ f(Λ(X)). Then there exists s1 ∈ Λ(X) such that

f(s1) =t1. Thus we getP SΛ(s1)⊆X. We shall prove that

P SΥ(t1)⊆f(X). Lett2∈P SΥ(t1). Then there exists2∈

S such thatf(s2) =t2. Thus we havef(s2)∈P SΥ(f(s1)).

By the argument (1), we obtain that s2 ∈ P SΛ(s1), and

so s2 ∈ X. Hence t2 = f(s2) ∈ f(X), and thus,

P SΥ(t1) ⊆ f(X). Therefore we have t1 ∈ Υ(f(X)). As

a consequence,f(Λ(X))⊆Υ(f(X)).

(4) Let X be a non-empty subset of S. We only need to prove that Υ(f(X)) ⊆ f(Λ(X)). Let t1 ∈ Υ(f(X)).

Then there exists s1 ∈ S such that f(s1) = t1. Thus

P SΥ(f(s1)) ⊆f(X). We shall show that P SΛ(s1) ⊆ X.

Let s2 ∈ P SΛ(s1). Then, by the argument (1), we have

f(s2) ∈ P SΥ(f(s1)). Hence f(s2) ∈ f(X). Thus there

existss3 ∈X such thatf(s3) =f(s2). By the assumption,

s2 ∈X, and so P SΛ(s1)⊆X. Hences1 ∈Λ(X), and so

t1=f(s1)∈f(Λ(X)). ThusΥ(f(X))⊆f(Λ(X)).

By the argument (3), we get f(Λ(X)) ⊆ Υ(f(X)). Consequently,f(Λ(X)) =Υ(f(X)).

(5) The proof is straightforward, so we omit it.

Proposition 11. Let f be an isomorphism from S in

(S,PSΛ(S))toT in(T,PSΥ(T)), where the binary relation

Λ := {(s1, s2) ∈ S ×S : (f(s1), f(s2)) ∈ Υ}. If Υ is

complete, thenΛ is complete.

Proof: Let s1, s2 be two elements in S. Suppose that

s3 ∈ P SΛ(s1s2). Then, by Proposition 10 (1), we get that

f(s3)∈P SΥ(f(s1s2)). Since f is a homomorphism andΥ

is complete, we have

f(s3)∈P SΥ(f(s1s2))

=P SΥ(f(s1)f(s2))

= (P SΥ(f(s1)))(P SΥ(f(s2))).

Thus there exist t1 ∈ P SΥ(f(s1)), t2 ∈ P SΥ(f(s2)) such

thatf(s3) =t1t2. Sincef is surjective, there exists4, s5∈S

such thatf(s4) =t1 andf(s5) =t2. Since

f(s4)f(s5) =f(s3)∈(P SΥ(f(s1)))(P SΥ(f(s2))),

we havef(s4)∈P SΥ(f(s1))andf(s5)∈P SΥ(f(s2)). By

Proposition 10 (1),s4∈P SΛ(s1)ands5∈P SΛ(s2). Sincef

is a homomorphism,f(s3) =f(s4)f(s5) =f(s4s5). By the

assumption, we obtain thats3=s4s5. Thus we get thats3∈

P SΛ(s1)P SΛ(s2). Therefore we obtain that P SΛ(s1s2) ⊆

P SΛ(s1)P SΛ(s2).

On the other hand, by Propositions 7 and 10 (5), we obtain that P SΛ(s1)P SΛ(s2) ⊆ P SΛ(s1s2). Thus

P SΛ(s1)P SΛ(s2) = P SΛ(s1s2). Hence we get PSΛ(S)is

Λ-complete. ThusΛ is complete.

Theorem 7. Let f be an epimorphism fromSin(S,PSΛ(S)) to T in(T,PSΥ(T))type PCR, where the binary relation

Λ := {(s1, s2) ∈ S ×S : (f(s1), f(s2)) ∈ Υ}. If X is

a non-empty subset of S and f is injective, then Λ(X) is an PSΛ(S)-upper approximation semigroup if and only if

Υ(f(X))is anPSΥ(T)-upper approximation semigroup.

Proof:Suppose thatΛ(X)is anPSΛ(S)-upper

approx-imation semigroup. Then, by Proposition 10 (2), we obtain that

(Υ(f(X)))(Υ(f(X))) =(f(Λ(X)))(f(Λ(X)))

=f((Λ(X))(Λ(X)))

⊆f(Λ(X)) =Υ(f(X)).

Hence Υ(f(X)) is a subsemigroup of T. Thus we get Υ(f(X))is anPSΥ(T)-upper approximation semigroup.

Conversely, we suppose thats1 ∈(Λ(X))(Λ(X)). From

Proposition 10 (2), it follows that

f(s1)∈f((Λ(X))(Λ(X)))

=(f(Λ(X)))(f(Λ(X)))

=(Υ(f(X)))(Υ(f(X)))

⊆Υ(f(X)) =f(Λ(X)).

Thus there exists s2 ∈ Λ(X) such that f(s1) = f(s2).

Hence we haveP SΛ(s2)∩X ̸=∅. By the assumption, we

obtain that P SΛ(s1)∩X ≠ ∅, and so s1 ∈ Λ(X). Hence

(Λ(X))(Λ(X)) ⊆ Λ(X). Thus Λ(X) is a subsemigroup of S. Therefore, Λ(X) is an PSΛ(S)-upper approximation

semigroup.

Theorem 8. Let f be an epimorphism fromSin(S,PSΛ(S)) to T in(T,PSΥ(T))type PCR, where the binary relation

Λ:={(s1, s2)∈S×S: (f(s1), f(s2))∈Υ}. IfXis a

non-empty subset ofSand f is injective, thenΛ(X)is aPSΛ(S) -lower approximation semigroup if and only ifΥ(f(X))is a

PSΥ(T)-lower approximation semigroup.

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Proof: By Proposition 10 (4) and using the similar method in the proof of Theorem 7, we can prove that the statement holds.

The following corollary is immediate consequences of Theorems 7 and 8.

Corollary 4. Let f be an epimorphism from S in

(S,PSΛ(S))toT in(T,PSΥ(T))type PCR, where the bi-nary relation Λ:={(s1, s2)∈S×S: (f(s1), f(s2))∈Υ}.

IfX is a non-empty subset ofSand f is injective, thenΛ(X)

is a PSΛ(S)-rough semigroup if and only ifΥ(f(X)) is a

PSΥ(T)-rough semigroup.

Theorem 9. Let f be an epimorphism fromSin(S,PSΛ(S)) to T in (T,PSΥ(T)) type PCR, where the binary relation

Λ := {(s1, s2) ∈ S ×S : (f(s1), f(s2)) ∈ Υ}. If X is a

non-empty subset of S and f is injective, then Λ(X) is an

PSΛ(S)-upper approximation ideal if and only ifΥ(f(X)) is an PSΥ(T)-upper approximation ideal.

Proof:Suppose thatΛ(X)is anPSΛ(S)-upper

approx-imation ideal. Then we have SΛ(X)⊆Λ(X). Whence we have f(SΛ(X)) ⊆ f(Λ(X)). By Proposition 10 (2), we obtain that

T Υ(f(X)) =f(SΛ(X))⊆f(Λ(X)) =Υ(f(X)).

HenceΥ(f(X))is a left ideal ofT. Similarly, we can prove that Υ(f(X)) is a right ideal of T. Thus Υ(f(X)) is an PSΥ(T)-upper approximation ideal.

Conversely, we suppose that Υ(f(X)) is an PSΥ(T)

-upper approximation ideal. Then, T Υ(f(X)) ⊆ Υ(f(X)). Now, let s1 ∈SΛ(X). From Proposition 10 (2), it follows

that

f(s1)∈f(SΛ(X)) =T Υ(f(X))⊆Υ(f(X)) =f(Λ(X)).

Thus there exists s2∈Λ(X) such that f(s1) = f(s2), and

so P SΛ(s2)∩X ̸= ∅. By the assumption, we have that

P SΛ(s1)∩X̸=∅, and sos1∈Λ(X). Thus we getSΛ(X)⊆

Λ(X). Then, Λ(X) is a left ideal of S. Similarly, we can check that Λ(X) is a right ideal of S. Therefore, Λ(X) is an PSΛ(S)-upper approximation ideal.

Theorem 10. Let f be an epimorphism from S in

If X is a non-empty subset of S and f is injective, then

Λ(X)is aPSΛ(S)-lower approximation ideal if and only if

Υ(f(X))is aPSΥ(T)-lower approximation ideal.

If X is a non-empty subset of S and f is injective, then

Λ(X)is aPSΛ(S)-rough ideal if and only if Υ(f(X))is a

PSΥ(T)-rough ideal.

(S,PSΛ(S))toTin(T,PSΥ(T))type CR, where the binary

relation Λ := {(s1, s2) ∈ S×S : (f(s1), f(s2))∈ Υ}. If

X is a non-empty subset ofS and f is injective, then Λ(X)

is anPSΛ(S)-upper approximation completely prime ideal if and only ifΥ(f(X))is anPSΥ(T)-upper approximation completely prime ideal.

Proof: Assume that Λ(X) is an PSΛ(S)-upper

ap-proximation completely prime ideal. Let t1, t2 ∈ T be

such that t1t2 ∈ Υ(f(X)). Thus there exist s1, s2 ∈ S

such that f(s1) = t1 and f(s2) = t2. Hence we have

P SΥ(f(s1)f(s2))∩f(X)̸=∅. Since Υ is complete,

(P SΥ(f(s1)))(P SΥ(f(s2)))∩f(X)̸=∅.

Then there exists f(s3)∈ P SΥ(f(s1)) and exists f(s4)∈

P SΥ(f(s2)) such that f(s3)f(s4) ∈ f(X), and so

f(s3s4) ∈ f(X). Then there exists s5 ∈ X such that

f(s3s4) = f(s5). By Proposition 10 (1), we obtain that

s3 ∈ P SΛ(s1) and s4 ∈ P SΛ(s2). From Proposition 7

and Proposition 10 (5), it follows that s3s4 ∈ P SΛ(s1s2).

By the assumption, we get that s5 = s3s4 = P SΛ(s1s2).

Thus P SΛ(s1s2)∩X ≠ ∅, and so s1s2 ∈ Λ(X). Since

Λ(X) is a completely prime ideal of S, we have that s1∈Λ(X)ors2∈Λ(X). Hence we havef(s1)∈f(Λ(X))

or f(s2) ∈ f(Λ(X)). From Proposition 10 (2), we get

f(s1) ∈ Υ(f(X)) or f(s2) ∈ Υ(f(X)), which yields

t1 ∈ Υ(f(X)) or t2 ∈ Υ(f(X)). Thus Υ(f(X)) is a

completely prime ideal of T. Therefore, Υ(f(X)) is an PSΥ(T)-upper approximation completely prime ideal.

Conversely, we suppose that Υ(f(X)) is an PSΛ(S)

-upper approximation completely prime ideal. Let s6, s7

be two elements in S such that s6s7 ∈ Λ(X). Then,

f(s6s7)∈f(Λ(X)). By Proposition 10 (2), we obtain that

f(s6)f(s7) =f(s6s7)∈f(Λ(X)) =Υ(f(X)).

Thus f(s6) ∈ Υ(f(X)) or f(s7) ∈ Υ(f(X)). Now, we

consider the following two cases.

Case 1. If f(s6) ∈ Υ(f(X)), then we have that

f(s6) ∈ f(Λ(X)) since Proposition 10 (2). Thus there

exists s8 ∈ Λ(X) such that f(s6) = f(s8). Whence we

get P SΛ(s8)∩X ̸=∅. Thus P SΛ(s6)∩X ̸=∅ since f is

injective. Therefore,s6∈Λ(X).

Case 2. If f(s7)∈Υ(f(X)), then s7 ∈Λ(X) since the

proof is similar to that the first case.

Consequently, Λ(X) is an PSΛ(S)-upper approximation

completely prime ideal.

(S,PSΛ(S))toTin(T,PSΥ(T))type CR, where the binary relation Λ := {(s1, s2) ∈ S×S : (f(s1), f(s2))∈ Υ}. If

X is a non-empty subset ofS and f is injective, then Λ(X)

is a PSΛ(S)-lower approximation completely prime ideal if and only if Υ(f(X)) is a PSΥ(T)-lower approximation completely prime ideal.

(S,PSΛ(S))toTin(T,PSΥ(T))type CR, where the binary relation Λ := {(s1, s2) ∈ S×S : (f(s1), f(s2))∈ Υ}. If

IAENG International Journal of Applied Mathematics, 49:1, IJAM_49_1_08

(9)

X is a non-empty subset of S and f is injective, thenΛ(X)

is a PSΛ(S)-rough completely prime ideal if and only if

Υ(f(X))is aPSΥ(T)-rough completely prime ideal.

VI. DISCUSSIONS ANDCONCLUSIONS

In this section we discuss approximation forms of this research and models in [1], [4], [7], [13], [23].

Firstly, concepts of generalizations of rough sets in general have been established as the following the diagram.

[1] - [23] - Our rough sets

Based on this point, if the equivalence property of a relation is put in the Mareay’s rough set [23], then the Mareay’s rough set is a generalization of the Pawlak’s rough set [1]. Moreover, if our rough set is considered under the single universe and the equal condition in Definition 5, then such the rough set is a generalization of the Mareay’s rough set.

Secondly, we discuss main results in a semigroup of this work (Sections IV and V), Kuroki [4], Xiao and Zhang [7], and Wang and Zhan [13] by using TABLES IV, V and VI below.

In the following TABLES IV, V and VI, the symbol X denotes two statements as the following.

(1) The sufficient condition (briefly, SC) of an upper ap-proximation semigroup (briefly, UAS) (resp. a lower approximation semigroup (briefly, LAS) and a rough semigroup (briefly, RS)) is provided in [4], [7], [13], or this work. Similarly, if sufficient conditions of an upper approximation ideal (briefly, UAI) (resp. a lower approximation ideal (briefly, LAI) and a rough ideal (briefly, RI)) and an upper approximation completely prime ideal (briefly, UAC) (resp. a lower approximation completely prime ideal (briefly, LAC) and a rough completely prime ideal (briefly, RC)) are provided. (2) The relationship between the UAS (resp. LAS and RS)

and the homomorphic image of the UAS (resp. LAS and RS) is demonstrated under homomorphism problems (briefly, HP) in [4], [7], [13], or this work. Similarly, if UAI (resp. LAI and RI) and UAC (resp. LAC and RC) are examined under HP.

TABLE IV

THE RESULTS OF UPPER APPROXIMATIONS IN SEMIGROUPS

[4] [7] [13] Our Model

UAS (SC) X X X

UAI (SC) X X X

UAC (SC) X X X

UAS (HP) X

UAI (HP) X X

UAC (HP) X X

TABLE V

THE RESULTS OF LOWER APPROXIMATIONS IN SEMIGROUPS

[4] [7] [13] Our Model

LAS (SC) X X X

LAI (SC) X X X

LAC (SC) X X X

LAS (HP) X

LAI (HP) X X

LAC (HP) X X

TABLE VI

THE RESULTS OF ROUGH SETS IN SEMIGROUPS

[4] [7] [13] Our Model

RS (SC) X

RI (SC) X

RC (SC) X

RS (HP) X

RI (HP) X

RC (HP) X

From TABLES IV, V and VI, we observe that sufficient conditions are completely obtained in this research (Section IV). Furthermore, connections under homomorphism prob-lems are entirely verified in this work (Section V).

From the Mareay’s rough set induced by a binary relation on the single universe, a generalization of the Mareay’s rough set was constructed in an approximation space based on portions of successor classes induced by a binary rela-tion between two universes, and a corresponding example was gave. Moreover, interesting algebraic properties were investigated. Under a preorder and compatible relation, ap-proximation processings in semigroups were applied from the novel generalized rough set. As discussed above, it indicates that sufficient conditions of rough semigroups, rough ideals and rough completely prime ideals are fully obtained, and associations under homomorphism problems are ideally checked. The novel generalized rough set can be applied in a semigroup. However, when we consider other algebraic systems, the corresponding issues need to be further examined.

ACKNOWLEDGMENT

The authors would like to exhibit their honest thanks to the anonymous referees for their considerable suggestions.

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